Problem 62 - Algebra A Combined Function

Question

The production supervisor at Alexandra's Office Products finds that it takes 3 hours to manufacture a particular office chair and 6 hours to manufacture an office desk. A total of 1200 hours is available to produce office chairs and desks of this style. The linear equation that models this situation is \(3 x+6 y=1200,\) where \(x\) represents the number of chairs produced and y the number of desks manufactured. Complete the ordered pair solution ( \(, 0\) ) of this equation. Describe the manufacturing situation that corresponds to this solution.

Step-by-Step Solution

Answer

The solution is (400, 0), meaning 400 chairs are produced and no desks.

Step 1: Understand the Problem

You need to find the value of \(x\) in the ordered pair \((x, 0)\) for the equation \(3x + 6y = 1200\). In the context of this problem, \(x\) is the number of chairs and \(y\) is the number of desks. In the ordered pair \((x, 0)\), \(y = 0\), which means no desks are produced.

Step 2: Substitute in the Equation

Substitute \(y = 0\) into the equation \(3x + 6y = 1200\). This simplifies the equation to \(3x = 1200\), since \(6 imes 0 = 0\).

Step 3: Solve for x

Divide both sides of the equation \(3x = 1200\) by 3 to isolate \(x\). This gives \(x = \frac{1200}{3}\).

Step 4: Perform the Division

Calculate \(x = \frac{1200}{3}\), which results in \(x = 400\).

Step 5: Interpret the Solution

The ordered pair solution is \((400, 0)\). This means that 400 chairs can be produced if no desks are manufactured under the given labor constraints.

Key Concepts

Ordered Pairs

An ordered pair is a pair of numbers used to locate a point on a coordinate plane. It is expressed in the form \((x, y)\), where \(x\) is the first number and represents the horizontal position, and \(y\) is the second number, representing the vertical position. In our exercise, the ordered pair is \((x, 0)\), which tells us the position along the horizontal axis where no office desks are produced.

Ordered pairs are pivotal when solving linear equations because they represent the solutions graphically. By plugging the values into the equation, you can determine the specific situation they describe.

• The first value \(x\) specifically indicates the output of chairs produced.
• The second value, \(y\), being zero, means no desks are manufactured.

Understanding ordered pairs helps us translate mathematical solutions into real-world contexts, like our chair and desk production scenario.

Systems of Equations

A system of equations involves solving two or more equations that have common variables. Solving these systems helps find the point where the equations intersect, representing a solution that satisfies all equations simultaneously.

In our context, we had only one equation to consider: \(3x + 6y = 1200\), but this equation is a part of a potential system if there were more constraints to consider. The equation is a model of the production process limited by hours.

• The equation tells us how labor hours are distributed between chairs and desks.
• It provides insight into managing resources by stating that any combination of chairs and desks must respect the total time constraint of 1200 hours.

If you had another equation, you could find solutions that satisfy both conditions, greatly aiding in planning and decision-making.

Problem Solving

Problem solving involves using logical and mathematical techniques to find answers to questions or scenarios presented. In algebra, we often use equations to model and solve these real-world problems.

To solve our chair and desk manufacturing question:

• You first need to understand what the problem is asking by interpreting the given equation-related values.
• Next, by substituting \(y = 0\), recognize that you are solving for the condition where no desks are produced.
• Finally, solve the remaining equation \(3x = 1200\) to find that \(x = 400\), representing 400 chairs.

By following these steps, problem solving becomes a systematic approach to deriving meaningful conclusions from the information provided.